Residual Life Function Research Articles

Additive mean residual life model with covariate measurement errors

In this paper, we consider the additive mean residual life model when a proportion of covariates cannot be observed accurately but repeated measurements on covariates are available. The number of repeated measurements is allowed to vary across different subjects. A nonparametric-correction method is developed for inference about regression parameters and the baseline mean residual life function. The resultant estimators are shown to be consistent and asymptotically normal, and consistent standard error estimators are also provided. Simulation studies are carried out to examine the performance of the proposed approach. We illustrate the method by application to data from an HIV clinical trial.

Fitting and testing the Marshall–Olkin extended Weibull model with randomly censored data

The random censorship model (RCM) is commonly used in biomedical science for modeling life distributions. The popular non-parametric Kaplan–Meier estimator and some semiparametric models such as Cox proportional hazard models are extensively discussed in the literature. In this paper, we propose to fit the RCM with the assumption that the actual life distribution and the censoring distribution have a proportional odds relationship. The parametric model is defined using Marshall–Olkin's extended Weibull distribution. We utilize the maximum-likelihood procedure to estimate model parameters, the survival distribution, the mean residual life function, and the hazard rate as well. The proportional odds assumption is also justified by the newly proposed bootstrap Komogorov–Smirnov type goodness-of-fit test. A simulation study on the MLE of model parameters and the median survival time is carried out to assess the finite sample performance of the model. Finally, we implement the proposed model on two real-life data sets.

Conditional quantile residual lifetime models for right censored data

Quantile residual lifetime function is a more comprehensive quantitative measure for residual lifetimes than the mean residual lifetime function. It also incorporates the median residual life function, which is less restrictive than the model based on the mean residual lifetime. In this study, we propose a semiparametric estimator of the conditional quantile residual lifetime under different covariate effects at a specified time point by the reinforcement of the auxiliary models. Two kind of test statistics are proposed to compare two quantile residual lifetimes at fixed time points. Asymptotic properties are also established and a revised bootstrap method is proposed to estimate the asymptotic variance of the estimator. Simulation studies are reported to assess the finite sample properties of the proposed estimator and the performance of test statistics in terms of type I error probabilities and powers at fixed time points. We also compare the proposed method with the method of Jung et al. (Biometrics 65:1203-1212, 2009) through simulation studies. The proposed methods are applied to HIV data and some interesting results are presented.

On the change points of mean residual life and failure rate functions for some generalized gamma type distributions

Mean residual life and failure rate functions are ubiquitously employed in reliability analysis. The term of useful period of lifetime distributions of bathtub-shaped failure rate functions is referred to the flat rigion of this function and has attracted authors and researchers in reliability, actuary, and survival analysis. In recent years, considering the change points of mean residual life and failure rate functions has been extensively utelized in determining the optimum burn-in time. In this paper we investigate the difference between the change points of failure rate and mean residual life functions of some generalized gamma type distributions due to the capability of these distributions in modeling various bathtub-shaped failure rate functions.

An Efficient Estimation of the Mean Residual Life Function with Length-Biased Right-Censored Data

The mean residual life (MRL) function for a lifetime random variableT0is one of the basic parameters of interest in survival analysis. In this paper, we propose a new estimator of the MRL function with length-biased right-censored data and evaluate its performance through a small Monte Carlo simulation study. The results of the simulations show that the proposed estimator outperforms the existing one referred to in Data and Model Setup Section in terms of Monte Carlo bias and mean square error, especially when the censoring rate is heavy. We also show that the proposed estimator converges in distribution under some conditions.

Reliability Analysis of the Proportional Mean Residual Life Order

The concept of mean residual life plays an important role in reliability and life testing. In this paper, we introduce and study a new stochastic order called proportional mean residual life order. Several characterizations and preservation properties of the new order under some reliability operations are discussed. As a consequence, a new class of life distributions is introduced on the basis of the anti-star-shaped property of the mean residual life function. We study some reliability properties and some characterizations of this class and provide some examples of interest in reliability.

Inference for the median residual life function in sequential multiple assignment randomized trials

In survival analysis, median residual lifetime is often used as a summary measure to assess treatment effectiveness; it is not clear, however, how such a quantity could be estimated for a given dynamic treatment regimen using data from sequential randomized clinical trials. We propose a method to estimate a dynamic treatment regimen-specific median residual life (MERL) function from sequential multiple assignment randomized trials. We present the MERL estimator, which is based on inverse probability weighting, as well as, two variance estimates for the MERL estimator. One variance estimate follows from Lunceford, Davidian and Tsiatis' 2002 survival function-based variance estimate and the other uses the sandwich estimator. The MERL estimator is evaluated, and its two variance estimates are compared through simulation studies, showing that the estimator and both variance estimates produce approximately unbiased results in large samples. To demonstrate our methods, the estimator has been applied to data from a sequentially randomized leukemia clinical trial.

On the Bayesian estimation of the weighted Lindley distribution

The weighted distributions provide a comprehensive understanding by adding flexibility in the existing standard distributions. In this article, we considered the weighted Lindley distribution which belongs to the class of the weighted distributions and investigated various its properties. Although, our main focus is the Bayesian analysis however, stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics derivations are obtained first time for the said distribution. Different types of loss functions are considered; the Bayes estimators and their respective posterior risks are computed and compared. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also analysed. The Lindley approximation and the importance sampling are described for estimation of parameters. A simulation study is designed to inspect the effect of sample size on the estimated parameters. A real-life application is also presented for the illustration purpose.

Semiparametric proportional mean residual life model with covariates missing at random

In this paper, we consider statistical inference for the proportional mean residual life model when some covariates are missing at random. Simple and augmented inverse probability-weighted estimating equations are used to obtain the estimators of the regression coefficients and baseline mean residual life function. The unknown non-missingness probability and some unknown conditional expectations are estimated by the kernel smoothing technique. We show that the simple inverse probability-weighted estimator with estimated non-missingness probability is more efficient than that with the true non-missingness probability, while the augmented inverse probability-weighted estimator with estimated non-missingness probability and that with the true non-missingness probability have the same efficiency. The asymptotic properties of all the proposed estimators are established. Extensive simulation studies are conducted to examine the finite sample performance of the proposed estimator. At last, the proposed method is applied to the mouse leukaemia data.

Variance residual life function based on double truncation

Since, most of the real observations in industrial and reliability studies, are left, right or doubly truncated data, studying the reliability concepts of the components of a system or a device based on conditional random variables, are important and usual. One of the important and applicable reliability concepts, that recently has gathered the attention of the researchers, is the variance residual life. In this paper, we try to study some of the reliability properties of the variance residual life based on doubly truncated data. Its monotonicity properties and relations with doubly truncated mean residual life and doubly truncated residual coefficient of variation are discussed. Furthermore, the lower (upper) bound for it under some conditions is obtained. We also discuss and find the similar results for discrete random ageing which its differences with the continuous case, are noticeable. Finally, some examples due to this subject are mentioned.

A note on estimation of the mean residual life function with left-truncated and right-censored data

This note focuses on estimating the mean residual life function with left-truncated and right-censored data. We show that the proposed estimator converges weakly to a Gaussian process. The performances of the estimator and its pointwise confidence intervals are illustrated through simulation studies.

Reversed variance residual life function and its properties in discrete lifetime models

PurposeIn reliability studies, interests in discrete failure data came relatively late in comparison to its continuous analogue. Also, discrete failure data arise in several common situations. So, in this paper the authors try to study some reliability concepts such as reversed variance and reversed mean residual life functions based on discrete lifetime random variable.Design/methodology/approachSupposed T be a non‐negative discrete random variable, then based on reversed residual random variable Tk*=(k−T|T≤k), some useful and applicable relations and bounds are achieved.FindingsIn this paper, the authors study the reversed variance residual life in discrete lifetime distributions, the results of which are not similar to the continuous case. Its relationship with reversed mean residual life and reversed residual coefficient of variation are obtained. Also, its monotonicity and the associated ageing classes of distributions are discussed. Some characterization results of the class of increasing reversed variance residual life, which is denoted by IRVR, are presented and the upper bound for reversed variance residual life under some conditions is obtained.Practical implicationsThere are many situations where a continuous time is inappropriate for describing the lifetime of devices and other systems. For example, the lifetime of many devices in industry, such as switches and mechanical tools, depends essentially on the number of times that they are turned on and off or the number of shocks they receive. In such cases, the time to failure is often more appropriately represented by the number of times they are used before they fail, which is a discrete random variable.Originality/valueAll the results based on discrete reversed residual lifetime, such as the relationships among reversed mean, variance and coefficient of variation residual lifetime and also their monotonicity ageing classes, are new.

On the Mean Residual Life Function and Stress and Strength Analysis under Different Loss Function for Lindley Distribution

Purpose. Mathematical properties of Lindley distribution are derived under different loss functions. These properties include mean residual life function, Lorenz curve, stress and strength characteristic, and their respective posterior risk via simulation scheme. Methodology. Bayesian approach is used for the reliability characteristics. Results are compared on the basis of posterior risk. Findings. Using prior information on the parameter of Lindley distribution, Bayes estimates for reliability characteristics are compared under different loss functions. Practical Implications. Since Lindley distribution is a mixture of gamma and exponential distribution, so Bayesian estimation of reliability characteristics will have a great implication in reliability theory. Originality. A real life application to waiting time data at the bank is also described for the developed procedures. This study is useful for researcher and practitioner in reliability theory.

Local Polynomial Fitting of the Mean Residual Life Function

The mean residual life (MRL) function is one of the most important, widely used reliability measures in practice. For example, it is used to design burn-in programs, plan spare provision, and formulate warranty policies. Parametric techniques, which rely on the assumption that the parametric form of the failure time is known, are usually employed in estimating the MRL function. However, this approach could lead to an inconsistent, inaccurate estimator of the MRL function if the assumption is violated. A nonparametric approach in such a setup provides a promising alternative. In this paper, we employ local polynomial regression with fixed design points accompanied by appropriate binning to construct several new estimators for the MRL function. The asymptotic unbiasedness and consistency of the these estimators are proven. We then bring in two popular bandwidth selection methods to select the bandwidth of the proposed MRL estimators. Finally, we evaluate the performance of the estimators using several simulated and real life examples. Results indicate that the proposed estimators perform well in estimating MRL functions, particularly MRL models with constant, bathtub-shaped, and upside-down bathtub-shaped MRL functions.

Reliability of a K-Out-of-n System Equipped With a Single Warm Standby Component

A k-out-of-n:G system consists of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> components, and operates if at least <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> of its components operate. Its reliability properties have been widely studied in the literature from different perspectives. This paper is concerned with the reliability analysis of a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -out-of- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> :G system equipped with a single warm standby unit. We obtain an explicit expression for the reliability function of the system for arbitrary lifetime distributions. Two different mean residual life functions are also studied for the system.

On moments of k-th record values from the linear exponential distribution

Explicit formulae for single, central, and product moments of k-th upper and lower record values from the linear exponential distribution are given. The limited moments and the mean residual life function are also investigated. Moreover, recurrence relations for moments of k-th upper and lower record values from this distribution are derived.

ON A CLASS OF GENERALIZED MARSHALL–OLKIN BIVARIATE DISTRIBUTIONS AND SOME RELIABILITY CHARACTERISTICS

We consider here a general class of bivariate distributions from reliability point of view, and refer to it as generalized Marshall–Olkin bivariate distributions. This class includes as special cases the Marshall–Olkin bivariate exponential distribution and the class of bivariate distributions studied recently by Sarhan and Balakrishnan [25]. For this class, the reliability, survival, hazard, and mean residual life functions are all derived, and their monotonicity is discussed for the marginal as well as the conditional distributions. These functions are also studied for the series and parallel systems based on this bivariate distribution. Finally, the Clayton association measure for this bivariate model is derived in terms of the hazard gradient.

A two-parameter Lindley distribution

A two-parameter Lindley distribution, of which the Lindley distribution (LD) is a particular case, has been introduced. Its moments, failure rate function, mean residual life function and stochastic orderings have been discussed. The maximum likelihood method and the method of moments have been discussed for estimating its parameters. The distribution has been fitted to some data-sets to test its goodness of fit.

Reliability and Mean Residual Life of Complex Systems With Two Dependent Components Per Element

The reliability and mean residual life of complex systems are discussed. These systems consist of n elements each having two s -dependent subcomponents. The reliability of such systems involves the distributions of bivariate order statistics, and are connected with a bivariate binomial distribution. The mean residual life function of complex systems with intact components at time t is also discussed. Some examples and graphical representations are given.

Bootstrap Kernel Estimator of Mean Residual Life Function

The bootstrap kernel estimator for the mean residual life function is constructed. Its asymptotic properties are analyzed.